A unit is presented for the transfer of a gaseous component in one stream to or from a liquid component in another stream, in which a membrane assembly is wound spirally around a central rigid cylindrical core. The membrane assembly comprises a continuous length of flattened tubular membrane with a gas net passing through the intermediate portion of the membrane from end to end thereof and substantially the width of the interior of the tubular membrane. In forming the coil, a net separator is wound between the membrane layers of the coil. An inlet tube for one of the components communicates with one end of the membrane and an outlet tube for the same component communicates with the other end of the membrane. Means is provided for passing another of the components longitudinally of the coil between the membrane layers from one end of the coil to the other end. The specific embodiment of a blood oxygenator is described.